TITLE: Easy to state applications of dimension theory in algebraic geometry
QUESTION [7 upvotes]: Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages. 
Shafarevich gives two nice applications of the theory. 1) A proof of Tzen's theorem 2) A proof of existence of a line on a cubic surface in $\mathbb P^3$. 
Both applications are geometric statements. I would like to learn about some other applications that are easy to state. I am asking this partially because of an introductory course in algebraic geometry that I am teaching. I would like to give some motivation for developing the dimension theory.

REPLY [5 votes]: This isn't a very direct answer, but one of the things I use all the time in algebraic geometry as opposed to non-algebraic is that if $X$ is dense in $Y$, then $\dim(Y\setminus X) < \dim X$. The topologist's sine curve is a counterexample to this in non-algebraic geometry.